Question

Consider the difference equation
xt+1 = xteλ−xt ,
where the system domain is X = [0, +∞) and where λ is a positive parameter.
(a) Determine all fixed points of this difference equation.
(b) Linearize the difference equation in each of its fixed points.
(c) Prove that the difference equation has a periodic point of period 2 whenever λ > 2

Answer 1

Before the birth of her baby, a mother opens a saving account with fixed annual interest rate r. On each birthday of the child the mother deposits 100 times as many Euros as the current age of the child. Between every two consecutive birthdays, the bank pays interest into the account.
Derive a difference equation that describes the evolution of the balance of the saving account. What is the correct initial value for this difference equation? Determine a closed form expression for the balance of the saving account with t denoting the age of the child.

Answer 2

i think the balance will look like this sum:
\[\large \sum_{n=1}^{t}100n(1+r)^{t-n}\]

Answer 3

thank you veru much

Answer 4

and can you please tell me what is "n " in that case

Answer 5

just a iterative variable that starts at 1 and stops at t(age of child)
lets say after childs 3rd birthday...t=3
\[B = 100(1+r)^{2}+200(1+r)+300\]

Answer 6

aha i undrestand
and what do you thing about initial value for this diff.equation

Answer 7

i would think it would be 100 on first b-day...t=1

Answer 8

ok thank you so much really

Answer 9

o sorry but i have one more question

Answer 10

what about that bank pays interest into the account between everz two years

Answer 11

how can i show that

Answer 12

thats the (1+r) part...every year the bank pays interest such that balance increases by factor of (1+r) where r is annual int rate

Answer 13

but the bank Pay between everz two birthdays, is tah the same?

Answer 14

i don't know...i just assumed it was every year...every year is between 2 years ?

Answer 15

hmmmm....do you thing that is true: B= 100+200(r+1)+300.
bank pays only for second birthday if we show for 3 birthdays?and in this case the mother put 100 for firs year.but how to find how mich mother put bevor the baby born?

Answer 16

before the baby born

Answer 17

no no the bank pay between every consecutive birthdays

Answer 18

how is that possible? you can't deposit negative amounts of money
babies 1st birthday is when first deposit is made

Answer 19

yes you are absolute right

Answer 20

do you know how to slove that
Consider the difference equation
xt+1 = xteλ−xt ,
where the system domain is X = [0, +∞) and where λ is a positive parameter.
(a) Determine all fixed points of this difference equation.
(b) Linearize the difference equation in each of its fixed points.
(c) Prove that the difference equation has a periodic point of period 2 whenever λ > 2

Answer 21

not sure what you mean...this looks like a geometric sequence
\[X_{t+1}=X_t(e^{\lambda}-1) \rightarrow X_t = X_1(e^{\lambda}-1)^{t-1}\]